
// Make newform 243.2.a.f in Magma, downloaded from the LMFDB on 08 May 2024.
// To make the character of type GrpDrchElt, type "MakeCharacter_243_a();"
// To make the coeffs of the qexp of the newform in the Hecke field type "qexpCoeffs();"
// To make the newform (type ModFrm), type "MakeNewformModFrm_243_2_a_f();".
// This may take a long time!  To see verbose output, uncomment the SetVerbose lines below.
// The precision argument determines an initial guess on how many Fourier coefficients to use.
// This guess is increased enough to uniquely determine the newform.
// To make the Hecke irreducible modular symbols subspace (type ModSym)
// containing the newform, type "MakeNewformModSym_243_2_a_f();".
// This may take a long time!  To see verbose output, uncomment the SetVerbose line below.
// The default sign is -1.  You can change this with the optional parameter "sign".
function ConvertToHeckeField(input: pass_field := false, Kf := [])
    if not pass_field then
        poly := [-1, -3, 0, 1];
        Kf := NumberField(Polynomial([elt : elt in poly]));
        AssignNames(~Kf, ["nu"]);
    end if;
    Rf_num := [[1, 0, 0], [0, 1, 0], [-2, 0, 1]];
    Rf_basisdens := [1, 1, 1];
    Rf_basisnums := ChangeUniverse([[z : z in elt] : elt in Rf_num], Kf);
    Rfbasis := [Rf_basisnums[i]/Rf_basisdens[i] : i in [1..Degree(Kf)]];
    inp_vec := Vector(Rfbasis)*ChangeRing(Transpose(Matrix([[elt : elt in row] : row in input])),Kf);
    return Eltseq(inp_vec);
end function;


// To make the character of type GrpDrchElt, type "MakeCharacter_243_a();"
function MakeCharacter_243_a()
    N := 243;
    order := 1;
    char_gens := [2];
    v := [1];
    // chi(gens[i]) = zeta^v[i]
    assert UnitGenerators(DirichletGroup(N)) eq char_gens;
    F := CyclotomicField(order);
    chi := DirichletCharacterFromValuesOnUnitGenerators(DirichletGroup(N,F),[F|F.1^e:e in v]);
    return MinimalBaseRingCharacter(chi);
end function;

function MakeCharacter_243_a_Hecke(Kf)
    return MakeCharacter_243_a();
end function;


function ExtendMultiplicatively(weight, aps, character)
    prec := NextPrime(NthPrime(#aps)) - 1; // we will able to figure out a_0 ... a_prec
    primes := PrimesUpTo(prec);
    prime_powers := primes;
    assert #primes eq #aps;
    log_prec := Floor(Log(prec)/Log(2)); // prec < 2^(log_prec+1)
    F := Universe(aps);
    FXY<X, Y> := PolynomialRing(F, 2);
    // 1/(1 - a_p T + p^(weight - 1) * char(p) T^2) = 1 + a_p T + a_{p^2} T^2 + ...
    R<T> := PowerSeriesRing(FXY : Precision := log_prec + 1);
    recursion := Coefficients(1/(1 - X*T + Y*T^2));
    coeffs := [F!0: i in [1..(prec+1)]];
    coeffs[1] := 1; //a_1
    for i := 1 to #primes do
        p := primes[i];
        coeffs[p] := aps[i];
        b := p^(weight - 1) * F!character(p);
        r := 2;
        p_power := p * p;
        //deals with powers of p
        while p_power le prec do
            Append(~prime_powers, p_power);
            coeffs[p_power] := Evaluate(recursion[r + 1], [aps[i], b]);
            p_power *:= p;
            r +:= 1;
        end while;    
    end for;
    Sort(~prime_powers);
    for pp in prime_powers do
        for k := 1 to Floor(prec/pp) do
            if GCD(k, pp) eq 1 then
                coeffs[pp*k] := coeffs[pp]*coeffs[k];
            end if;
        end for;
    end for;
    return coeffs;
end function;


function qexpCoeffs()
    // To make the coeffs of the qexp of the newform in the Hecke field type "qexpCoeffs();"
    weight := 2;
    raw_aps := [[1, -1, 0], [0, 0, 0], [2, 1, 0], [-1, 1, -2], [1, 2, -3], [-1, 1, 1], [3, 0, 0], [-1, 0, 3], [2, -2, 3], [4, -1, 3], [-4, 1, -2], [-1, -3, 0], [-1, -5, 3], [-4, -2, 1], [-2, 5, 0], [6, 0, -3], [-7, 1, 0], [2, 1, -5], [2, -5, 1], [-3, -9, 3], [2, 6, -3], [2, -5, 4], [-2, -1, -3], [0, 9, -6], [5, -8, 4], [4, -4, -3], [5, 4, 4], [3, 0, 3], [-10, -3, 0], [5, 1, -6], [-1, 3, -3], [-4, 7, 6], [7, -1, 6], [5, 7, -5], [5, 4, -9], [8, 4, -5], [-7, -11, 10], [-1, 3, -12], [-1, 1, -3], [7, -7, 3], [-3, 0, -6], [-1, -6, -3], [-11, -1, -3], [-10, 1, 4], [12, 0, 6], [-1, -6, 6], [-7, 1, 4], [-1, 7, 4], [-11, 2, 3], [-10, 10, -8], [3, 9, -9], [-16, -2, -6], [2, 10, 1], [-15, -9, 6], [-1, -5, -6], [13, 5, -3], [9, 0, 9], [-7, -3, 6], [-1, 13, -17], [-2, -16, 12], [-1, 7, 7], [8, 16, -9], [11, -12, -3], [2, 4, 6], [5, -2, 1], [25, -4, 0], [-1, -20, 4], [8, -8, 10], [11, 7, -6], [20, 4, 4], [-17, 8, -9], [15, 9, -9], [-19, 4, -5], [11, -5, -11], [-1, 18, -15], [5, 10, 3], [4, 11, -9], [8, -9, -6], [-4, -2, 6], [-4, 1, 7], [14, -14, 0], [-4, -5, 16], [3, -9, -15], [-19, -15, 6], [2, -14, 16], [16, -7, 12], [-18, 9, -21], [-7, -2, 7], [13, -10, -6], [5, -5, -17], [21, -9, -3], [-14, -13, 15], [8, 9, 3], [8, -8, 6], [2, 7, -5], [24, 0, 3], [-4, -11, 15], [-27, 0, 3], [2, -12, 15], [17, -3, 18], [-16, -5, 7], [15, -9, -6], [14, -11, 15], [-2, 8, 15], [23, -2, 13], [2, 15, -12], [-17, -22, 12], [6, -9, -12], [-1, 1, 6], [-19, -2, -5], [-10, -20, 16], [11, -3, -3], [-7, -11, 12], [-13, 7, -23], [8, 27, -15], [-5, 14, -12], [-13, 10, -17], [3, 9, -6], [23, 19, -15], [1, 11, 6], [2, 10, -20], [-16, 7, -11], [1, -1, -18], [-9, 9, 3], [-13, 19, 13], [15, 0, 24], [-1, -2, 13], [-21, 0, 12], [14, -8, -2], [-13, -2, -14], [-19, -15, 15], [23, -14, 0], [-25, 10, 13], [-25, 15, 6], [-25, 10, 9], [-10, 16, -11], [0, -27, 33], [8, -20, 19], [-16, 4, 15], [-18, 9, -9], [17, 0, 3], [13, -16, 9], [17, -5, 4], [-9, 18, -3], [8, 24, -18], [4, -10, 21], [17, 7, -11], [-17, -4, 3], [29, 10, -14], [3, 9, 12], [-10, -5, -5], [3, 0, 18], [-1, 24, 0], [-1, -11, 9], [-4, -8, 4], [-2, 5, 9], [11, 6, 6], [-23, 11, 0], [8, -15, 6], [-22, 16, -21], [10, -19, 9], [-3, -18, 0], [23, 13, 4], [24, 18, -3], [-16, 1, 18], [-14, 2, 0], [2, 0, 0], [5, -5, 28]];
    aps := ConvertToHeckeField(raw_aps);
    chi := MakeCharacter_243_a_Hecke(Universe(aps));
    return ExtendMultiplicatively(weight, aps, chi);
end function;


// To make the newform (type ModFrm), type "MakeNewformModFrm_243_2_a_f();".
// This may take a long time!  To see verbose output, uncomment the SetVerbose lines below.
// The precision argument determines an initial guess on how many Fourier coefficients to use.
// This guess is increased enough to uniquely determine the newform.
function MakeNewformModFrm_243_2_a_f(:prec:=3)
    chi := MakeCharacter_243_a();
    f_vec := qexpCoeffs();
    Kf := Universe(f_vec);
    // SetVerbose("ModularForms", true);
    // SetVerbose("ModularSymbols", true);
    S := CuspidalSubspace(ModularForms(chi, 2));
    S := BaseChange(S, Kf);
    maxprec := NextPrime(997) - 1;
    while true do
        trunc_vec := Vector(Kf, [0] cat [f_vec[i]: i in [1..prec]]);
        B := Basis(S, prec + 1);
        S_basismat := Matrix([AbsEltseq(g): g in B]);
        if Rank(S_basismat) eq Min(NumberOfRows(S_basismat), NumberOfColumns(S_basismat)) then
            S_basismat := ChangeRing(S_basismat,Kf);
            f_lincom := Solution(S_basismat,trunc_vec);
            f := &+[f_lincom[i]*Basis(S)[i] : i in [1..#Basis(S)]];
            return f;
        end if;
        error if prec eq maxprec, "Unable to distinguish newform within newspace";
        prec := Min(Ceiling(1.25 * prec), maxprec);
    end while;
end function;


// To make the Hecke irreducible modular symbols subspace (type ModSym)
// containing the newform, type "MakeNewformModSym_243_2_a_f();".
// This may take a long time!  To see verbose output, uncomment the SetVerbose line below.
// The default sign is -1.  You can change this with the optional parameter "sign".
function MakeNewformModSym_243_2_a_f( : sign := -1)
    R<x> := PolynomialRing(Rationals());
    chi := MakeCharacter_243_a();
    // SetVerbose("ModularSymbols", true);
    Snew := NewSubspace(CuspidalSubspace(ModularSymbols(chi,2,sign)));
    Vf := Kernel([<2,R![3, 0, -3, 1]>,<7,R![-17, -6, 3, 1]>],Snew);
    return Vf;
end function;